A random variable let say X is taken as a function whose domain is the S i.e. sample space, and then the range is being set.

A sample space consisting of discrete elements then the r. is known as the discrete r.v. If the sample space consists of uninterrupted elements, then it is called as continuous r.v. The probability distribution is the distribution of the random variables. On the other hand, regarding r.v., the probability distribution is called as the discrete distribution or the continuous distribution. Thus the discrete distribution is thus represented by probability mass function given below:

 x -1 0 1 p(x) 0.2 0.4 0.4

Thus it is a discrete distribution. Here, the variable X contains values like -1, 0 and 1 with probability 0.2, 0.4 and 0.4 respectively.

So, the characteristic of pmf or the probable mass function are as follows:

1. P(x) 2: 0 for all x
2. Σ p(x) = 1

Any continuous distribution is represented by probability density function or pdf. Let’s consider an example,

f(x) = 1,                                0<x<1=0

Here the random variable can take any value between 0 and 1 with probability 1, and 0 for any other value.

Therefore the characteristic of pdf is:

1. F(x) ≥ 0 for all x
2. ʃ-∞ f(x) dx = 1  